论文标题
量子计算的可划分代码
Divisible Codes for Quantum Computation
论文作者
论文摘要
可划分的代码由代码文权重的属性定义，共享一个大于一个的共同除数。它们用于设计用于通信和传感的信号，本文探讨了它们如何用于保护量子信息，因为量子信息通过逻辑门转换。给定一个CSS代码$ \ MATHCAL {C} $，我们得出了对于横向对角线物理运算符$ u_z $保留$ \ Mathcal {C} $并引起$ u_l $所必需和足够的条件。 CSS代码$ \ MATHCAL {C} $中的$ z $ -Stabilizers由经典$ [n，k_1] $ binary Code $ \ Mathcal {C} _1 $确定，并由$ x $ -Stabilizers确定，由经典的$ [n，k_2] $ binary Code确定在$ \ Mathcal {C} _1 $中。对角物理运算符$ u_z $修复CSS代码$ \ MATHCAL {C} $的要求导致对$ \ Mathcal {C} _2 $的权重的一致性的约束。这些约束非常适合可分配的代码，并代表了利用两到三个权重的古典代码的广泛文献。我们使用由二次形式定义的第一阶reed muller代码的ZS代码构建新的CSS代码家族。我们提供了一种可能具有独立关注的标准方法的简单替代方法（基于Dickson正常形式）。最后，我们开发了一种绕过Eastin-Knill定理的方法，该方法指出，QECC不能仅通过横向门实现一套通用的逻辑大门。必不可少的想法是设计稳定器代码，其中$ n_1 $内盘和$ n_2 $的外Qubits，并在内部量子器上组装一套通用的故障耐受门。
Divisible codes are defined by the property that codeword weights share a common divisor greater than one. They are used to design signals for communications and sensing, and this paper explores how they can be used to protect quantum information as it is transformed by logical gates. Given a CSS code $\mathcal{C}$, we derive conditions that are both necessary and sufficient for a transversal diagonal physical operator $U_Z$ to preserve $\mathcal{C}$ and induce $U_L$. The group of $Z$-stabilizers in a CSS code $\mathcal{C}$ is determined by the dual of a classical $[n, k_1]$ binary code $\mathcal{C}_1$, and the group of $X$-stabilizers is determined by a classical $[n, k_2]$ binary code $\mathcal{C}_2$ that is contained in $\mathcal{C}_1$. The requirement that a diagonal physical operator $U_Z$ fixes a CSS code $\mathcal{C}$ leads to constraints on the congruence of weights in cosets of $\mathcal{C}_2$. These constraints are a perfect fit to divisible codes, and represent an opportunity to take advantage of the extensive literature on classical codes with two or three weights. We construct new families of CSS codes using cosets of the first order Reed Muller code defined by quadratic forms. We provide a simple alternative to the standard method of deriving the coset weight distributions (based on Dickson normal form) that may be of independent interest. Finally, we develop an approach to circumventing the Eastin-Knill Theorem which states that no QECC can implement a universal set of logical gates through transversal gates alone. The essential idea is to design stabilizer codes in layers, with $N_1$ inner qubits and $N_2$ outer qubits, and to assemble a universal set of fault tolerant gates on the inner qubits.
